scholarly journals Two-loop leading colour QCD corrections to $$ q\overline{q} $$ → γγg and qg → γγq

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Bakul Agarwal ◽  
Federico Buccioni ◽  
Andreas von Manteuffel ◽  
Lorenzo Tancredi
Keyword(s):  
Numerical Code ◽  
Tree Level ◽  
Partial Fraction ◽  
Integration By Parts ◽  
Analytic Expressions ◽  
Partial Fraction Decomposition ◽  
Loop Amplitudes

Abstract We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

scholarly journals Loops in AdS: from the spectral representation to position space. Part II

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Dean Carmi
Keyword(s):  
Spectral Representation ◽  
Tree Level ◽  
Point Function ◽  
Position Space ◽  
Gross Neveu Model ◽  
Loop Amplitudes

Abstract We continue the study of AdS loop amplitudes in the spectral representation and in position space. We compute the finite coupling 4-point function in position space for the large-N conformal Gross Neveu model on AdS3. The resummation of loop bubble diagrams gives a result proportional to a tree-level contact diagram. We show that certain families of fermionic Witten diagrams can be easily computed from their companion scalar diagrams. Thus, many of the results and identities of [1] are extended to the case of external fermions. We derive a spectral representation for ladder diagrams in AdS. Finally, we compute various bulk 2-point correlators, extending the results of [1].

scholarly journals Rational Solutions of Riccati Differential Equation with Coefficients Rational

2011 ◽  
Vol 2011 ◽  
pp. 1-44
Author(s):  
Nadhem Echi
Keyword(s):  
Differential Equation ◽  
Galois Group ◽  
Efficient Method ◽  
Rational Solution ◽  
Partial Fraction ◽  
Rational Solutions ◽  
Riccati Differential Equation ◽  
Differential Galois Group ◽  
Partial Fraction Decomposition ◽  
Decomposition Of Solution

This paper presents a simple and efficient method for determining the rational solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation , is reducible, we look for the rational solutions of Riccati differential equation , by reducing the number of checks to be made and by accelerating the search for the partial fraction decomposition of the solution reserved for the poles of which are false poles of . This partial fraction decomposition of solution can be used to code . The examples demonstrate the effectiveness of the method.

Parallel Solution of Linear Systems

2016 ◽  
Vol 6 (3) ◽  
pp. 278-289
Author(s):  
Sidi-Mahmoud Kaber ◽  
Amine Loumi ◽  
Philippe Parnaudeau
Keyword(s):  
Linear Systems ◽  
Numerical Algorithms ◽  
Parallel Architecture ◽  
Decomposition Methods ◽  
Partial Fraction ◽  
Domain Decomposition Methods ◽  
Simple Method ◽  
Parallel Solution ◽  
Partial Fraction Decomposition ◽  
Programming Paradigms

AbstractComputational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.

scholarly journals Auotomodeling rational mnemofunctions and their link to an analytical representation of distributions

2019 ◽  
Vol 55 (3) ◽  
pp. 288-298
Author(s):  
T. R. Shahava
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Real Line ◽  
Analytical Representation ◽  
Partial Fraction ◽  
Boundary Values ◽  
Multiplication Rule ◽  
Space Of Distributions ◽  
Partial Fraction Decomposition ◽  
Proper Rational Function

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

scholarly journals Fast Algorithms for Partial Fraction Decomposition

1977 ◽  
Vol 6 (3) ◽  
pp. 582-593 ◽  
Author(s):  
H. T. Kung ◽  
D. M. Tong
Keyword(s):  
Fast Algorithms ◽  
Partial Fraction ◽  
Partial Fraction Decomposition
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